Generalized flows, intrinsic stochasticity, and turbulent transport

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows, which are families of probability distributions on the space of solutions to the associated ordinary differential equations which no longer satisfy the uniqueness theorem for ordinary differential equations. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field, are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes, characterized by their degrees of compressibility, are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behaviors for the structure functions of the passive scalar. Physically, this means that the scaling depends on Prandtl number. In the other two regimes, the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The “one force, one solution” principle is established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime, which is the regime of inverse cascade. Existence and uniqueness of an invariant measure are also proved in these regimes when the transport equation is suitably forced. Finally incomplete self similarity in the sense of Barenblatt and Chorin is established.

Predicting species diversity in tropical forests

A fundamental question in ecology is how many species occur within a given area. Despite the complexity and diversity of different ecosystems, there exists a surprisingly simple, approximate answer: the number of species is proportional to the size of the area raised to some exponent. The exponent often turns out to be roughly 1/4. This power law can be derived from assumptions about the relative abundances of species or from notions of self-similarity. Here we analyze the largest existing data set of location-mapped species: over one million, individually identified trees from five tropical forests on three continents. Although the power law is a reasonable, zeroth-order approximation of our data, we find consistent deviations from it on all spatial scales. Furthermore, tropical forests are not self-similar at areas ≤50 hectares. We develop an extended model of the species-area relationship, which enables us to predict large-scale species diversity from small-scale data samples more accurately than any other available method.

Scale dependence in earthquake phenomena and its relevance to earthquake prediction.

The recent discovery of a low-velocity, low-Q zone with a width of 50-200 m reaching to the top of the ductile part of the crust, by observations on seismic guided waves trapped in the fault zone of the Landers earthquake of 1992, and its identification with the shear zone inferred from the distribution of tension cracks observed on the surface support the existence of a characteristic scale length of the order of 100 m affecting various earthquake phenomena in southern California, as evidenced earlier by the kink in the magnitude-frequency relation at about M3, the constant corner frequency for earthquakes with M below about 3, and the sourcecontrolled fmax of 5-10 Hz for major earthquakes. The temporal correlation between coda Q-1 and the fractional rate of occurrence of earthquakes in the magnitude range 3-3.5, the geographical similarity of coda Q-1 and seismic velocity at a depth of 20 km, and the simultaneous change of coda Q-1 and conductivity at the lower crust support the hypotheses that coda Q-1 may represent the activity of creep fracture in the ductile part of the lithosphere occurring over cracks with a characteristic size of the order of 100 m. The existence of such a characteristic scale length cannot be consistent with the overall self-similarity of earthquakes unless we postulate a discrete hierarchy of such characteristic scale lengths. The discrete hierarchy of characteristic scale lengths is consistent with recently observed logarithmic periodicity in precursory seismicity.

Striking sequence similarity in inter- and intra-specific comparisons of class I SLG alleles from Brassica oleracea and Brassica campestris: Implications for the evolution and recognition mechanism

Self-incompatibility in Brassica is controlled by a single multi-allelic locus (S locus), which contains at least two highly polymorphic genes expressed in the stigma: an S glycoprotein gene (SLG) and an S receptor kinase gene (SRK). The putative ligand-binding domain of SRK exhibits high homology to the secretory protein SLG, and it is believed that SLG and SRK form an active receptor kinase complex with a self-pollen ligand, which leads to the rejection of self-pollen. Here, we report 31 novel SLG sequences of Brassica oleracea and Brassica campestris. Sequence comparisons of a large number of SLG alleles and SLG-related genes revealed the following points. (i) The striking sequence similarity observed in an inter-specific comparison (95.6% identity between SLG14 of B. oleracea and SLG25 of B. campestris in deduced amino acid sequence) suggests that SLG diversification predates speciation. (ii) A perfect match of the sequences in hypervariable regions, which are thought to determine S specificity in an intra-specific comparison (SLG8 and SLG46 of B. campestris) and the observation that the hypervariable regions of SLG and SRK of the same S haplotype were not necessarily highly similar suggests that SLG and SRK bind different sites of the pollen ligand and that they together determine S specificity. (iii) Comparison of the hypervariable regions of SLG alleles suggests that intragenic recombination, together with point mutations, has contributed to the generation of the high level of sequence variation in SLG alleles. Models for the evolution of SLG/SRK are presented.